On Hard Instances of Non-Commutative Permanent

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9797)

Abstract

Recent developments on the complexity of the non-commutative determinant and permanent [Chien et al. STOC 2011, Bläser ICALP 2013, Gentry CCC 2014] have settled the complexity of non-commutative determinant with respect to the structure of the underlying algebra. Continuing the research further, we look to obtain more insights on hard instances of non-commutative permanent and determinant.

We show that any Algebraic Branching Program (ABP) computing the Cayley permanent of a collection of disjoint directed two-cycles with distinct variables as edge labels requires exponential size. For graphs where every connected component contains at most six vertices, we show that evaluating the Cayley permanent over any algebra containing \(2\times 2\) matrices is \(\#\mathsf{P}\) complete.

Further, we obtain efficient algorithms for computing the Cayley permanent/determinant on graphs with bounded component size, when vertices within each component are not far apart from each other in the Cayley ordering. This gives a tight upper and lower bound for size of ABPs computing the permanent of disjoint two-cycles. Finally, we exhibit more families of non-commutative polynomial evaluation problems that are complete for \(\#\mathsf{P}\).

Our results demonstrate that apart from the structure of underlying algebras, relative ordering of the variables plays a crucial role in determining the complexity of non-commutative polynomials.

References

  1. 1.
    Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge University Press, Cambridge (2009)CrossRefMATHGoogle Scholar
  2. 2.
    Arvind, V., Joglekar, P.S., Srinivasan, S.: Arithmetic circuits and the hadamard product of polynomials. In: FSTTCS, pp. 25–36 (2009)Google Scholar
  3. 3.
    Arvind, V., Srinivasan, S.: On the hardness of the noncommutative determinant. In: STOC, pp. 677–686 (2010)Google Scholar
  4. 4.
    Aslaksen, H.: Quaternionic determinants. Math. Int. 18(3), 57–65 (1996)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Barvinok, A.I.: Two algorithmic results for the traveling salesman problem. Math. Oper. Res. 21(1), 65–84 (1996)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bläser, M.: Noncommutativity makes determinants hard. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part I. LNCS, vol. 7965, pp. 172–183. Springer, Heidelberg (2013)Google Scholar
  7. 7.
    Bürgisser, P.: Completeness and Reduction in Algebraic Complexity Theory. Springer, Heidelberg (2000)CrossRefMATHGoogle Scholar
  8. 8.
    Chien, S., Sinclair, A.: Algebras with polynomial identities and computing the determinant. SIAM J. Comput. 37(1), 252–266 (2007)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Datta, S., Kulkarni, R., Limaye, N., Mahajan, M.: Planarity, determinants, permanents, and (unique) matchings. ToCT 1(3), 10 (2010)CrossRefMATHGoogle Scholar
  10. 10.
    Engels, C., Raghavendra Rao, B.V.: New Algorithms and Hard Instances for Non-Commutative Computation. ArXiv e-prints, September 2014Google Scholar
  11. 11.
    Flarup, U., Koiran, P., Lyaudet, L.: On the expressive power of planar perfect matching and permanents of bounded treewidth matrices. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 124–136. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Flarup, U., Lyaudet, L.: On the expressive power of permanents and perfect matchings of matrices of bounded pathwidth/cliquewidth. ToCS 46(4), 761–791 (2010)MathSciNetMATHGoogle Scholar
  13. 13.
    Gentry, C.: Noncommutative determinant is hard: a simple proof using an extension of barrington’s theorem. In: CCC, pp. 181–187, June 2014Google Scholar
  14. 14.
    Limaye, N., Malod, G., Srinivasan, S.: Lower bounds for non-commutative skew circuits. In: Electronic Colloquium on Computational Complexity (ECCC), vol. 22, p. 22 (2015)Google Scholar
  15. 15.
    Mahajan, M., Rao, B.V.R.: Small space analogues of valiant’s classes and the limitations of skew formulas. Comput. Complex. 22(1), 1–38 (2013)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Nisan, N.: Lower bounds for non-commutative computation (extended abstract). In: STOC, pp. 410–418 (1991)Google Scholar
  17. 17.
    Shpilka, A., Yehudayoff, A.: Arithmetic circuits: a survey of recent results and open questions. FTTS 5(3–4), 207–388 (2010)MathSciNetMATHGoogle Scholar
  18. 18.
    Valiant, L.G.: Completeness classes in algebra. In: STOC 1979, pp. 249–261 (1979)Google Scholar
  19. 19.
    von zur Gathen, J.: Feasible arithmetic computations: Valiant’s hypothesis. J. Symb. Comput. 4(2), 137–172 (1987)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Tokyo Institute of TechnologyTokyoJapan
  2. 2.IIT MadrasChennaiIndia

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